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tensor-product finite element; local superconvergence; discrete Green's function
Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green's function and discrete derivative Green's function, and the relationship of norms in the finite element space such as $L^2$-norms, $W^{1,\infty }$-norms, and negative-norms in locally smooth subsets of the domain $\Omega $, locally pointwise superconvergence occurs in function values and derivatives.
[1] Brandts, J., Křížek, M.: History and future of superconvergence in three-dimensional finite element methods. Finite Element Methods. Three-Dimensional Problems P. Neittaanmäki, M. Křížek GAKUTO International Series. Mathematical Science Applications 15, Gakkotosho, Tokyo (2001), 22-33. MR 1896264 | Zbl 0994.65114
[2] Brandts, J., Křížek, M.: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23 (2003), 489-505. DOI 10.1093/imanum/23.3.489 | MR 1987941 | Zbl 1042.65081
[3] Brandts, J., Křížek, M.: Superconvergence of tetrahedral quadratic finite elements. J. Comput. Math. 23 (2005), 27-36. MR 2124141 | Zbl 1072.65137
[4] Chen, C. M.: Optimal points of stresses for the linear tetrahedral element. Nat. Sci. J. Xiangtan Univ. 3 (1980), 16-24 Chinese.
[5] Chen, C. M.: Construction Theory of Superconvergence of Finite Elements. Hunan Science and Technology Press, Changsha (2001), Chinese.
[6] Chen, L.: Superconvergence of tetrahedral linear finite elements. Int. J. Numer. Anal. Model. 3 (2006), 273-282. MR 2237882 | Zbl 1100.65084
[7] J. Douglas, Jr., T. Dupont, M. F. Wheeler: An $L^{\infty }$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials. Rev. Franc. Automat. Inform. Rech. Operat., Analyse Numer. 8 (1974), 61-66. DOI 10.1051/m2an/197408r200611 | MR 0359358 | Zbl 0315.65062
[8] Goodsell, G.: Gradient Superconvergence for Piecewise Linear Tetrahedral Finite Elements. Technical Report RAL-90-031, Science and Engineering Research Council, Rutherford Appleton Laboratory (1990).
[9] Goodsell, G.: Pointwise superconvergence of the gradient for the linear tetrahedral element. Numer. Methods Partial Differ. Equations 10 (1994), 651-666. DOI 10.1002/num.1690100511 | MR 1290950 | Zbl 0807.65112
[10] Hannukainen, A., Korotov, S., Křížek, M.: Nodal ${\mathcal O}(h^4)$-superconvergence in 3D by averaging piecewise linear, bilinear, and trilinear FE approximations. J. Comput. Math. 28 (2010), 1-10. DOI 10.4208/jcm.2009.09-m1004 | MR 2603577 | Zbl 1224.65247
[11] He, W. M., Guan, X. F., Cui, J. Z.: The local superconvergence of the trilinear element for the three-dimensional Poisson problem. J. Math. Anal. Appl. 388 (2012), 863-872. DOI 10.1016/j.jmaa.2011.10.031 | MR 2869793 | Zbl 1237.65115
[12] Kantchev, V., Lazarov, R.: Superconvergence of the gradient of linear finite elements for 3D Poisson equation. Proc. Int. Symp. Optimal Algorithms B. Sendov Bulgarian Academy of Sciences, Sofia (1986), 172-182. MR 1171706 | Zbl 0672.65088
[13] Lin, Q., Yan, N. N.: Construction and Analysis of High Efficient Finite Elements. Hebei University Press, Baoding (1996), Chinese.
[14] Lin, R., Zhang, Z.: Natural superconvergence points in three-dimensional finite elements. SIAM J. Numer. Anal. 46 (2008), 1281-1297. DOI 10.1137/070681168 | MR 2390994 | Zbl 1168.65059
[15] Liu, J., Jia, B., Zhu, Q.: An estimate for the three-dimensional discrete Green's function and applications. J. Math. Anal. Appl. 370 (2010), 350-363. DOI 10.1016/j.jmaa.2010.05.002 | MR 2651658 | Zbl 1194.35013
[16] Liu, J., Sun, H., Zhu, Q.: Superconvergence of tricubic block finite elements. Sci. China, Ser. A 52 (2009), 959-972. DOI 10.1007/s11425-009-0039-1 | MR 2505002 | Zbl 1183.65145
[17] Liu, J., Zhu, Q.: Estimate for the $W^{1,1}$-seminorm of discrete derivative Green's function in three dimensions. J. Hunan Univ. Arts Sci., Nat. Sci. 16 (2004), 1-3 Chinese. MR 2139634 | Zbl 1134.35344
[18] Liu, J., Zhu, Q.: Maximum-norm superapproximation of the gradient for quadratic finite elements in three dimensions. Acta Math. Sci., Ser. A, Chin. Ed. 26 (2006), 458-466 Chinese. MR 2243664 | Zbl 1154.65371
[19] Liu, J., Zhu, Q.: Pointwise supercloseness of tensor-product block finite elements. Numer. Methods Partial Differ. Equations 25 (2009), 990-1008. DOI 10.1002/num.20384 | MR 2526993 | Zbl 1178.65137
[20] Liu, J., Zhu, Q.: Pointwise supercloseness of pentahedral finite elements. Numer. Methods Partial Differ. Equations 26 (2010), 1572-1580. DOI 10.1002/num.20510 | MR 2732397 | Zbl 1204.65135
[21] Pehlivanov, A.: Superconvergence of the gradient for quadratic 3D simplex finite elements. Proceedings of the Conference on Numerical Methods and Application Bulgarian Academy of Sciences, Sofia (1989), 362-366. MR 1027639
[22] Schatz, A. H., Sloan, I. H., Wahlbin, L. B.: Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point. SIAM J. Numer. Anal. 33 (1996), 505-521. DOI 10.1137/0733027 | MR 1388486 | Zbl 0855.65115
[23] Zhang, Z., Lin, R.: Locating natural superconvergent points of finite element methods in 3D. Int. J. Numer. Anal. Model. 2 (2005), 19-30. MR 2112655 | Zbl 1071.65140
[24] Zhu, Q., Lin, Q.: Superconvergence Theory of the Finite Element Methods. Hunan Science and Technology Press, Changsha (1989), Chinese. MR 1200243
[25] Zlámal, M.: Superconvergence and reduced integration in the finite element method. Math. Comput. 32 (1978), 663-685. DOI 10.2307/2006479 | MR 0495027 | Zbl 0448.65068
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